Method of treating a subterranean formation using a rheology model for fluid optimization

ABSTRACT

Subterranean formation treatment methods incorporating a rheology model which enables prediction of fluid rheology properties during a treatment operation, where the foundation of the model is a description of the reaction chemistry which describes how the number of crosslinks and broken polymer linkages develops in time under the influence of crosslinkers, breakers, and/or thermally induced effects and pressure effects. In one aspect, when used as a tool for simulating the fluid viscosity, the model can help optimizing the fluid design and optional breaker schedule for a hydraulic fracturing treatment.

RELATED APPLICATION DATA

This application claims priority as a continuation application of U.S.patent application Ser. No. 11/625,105, filed Jan. 19, 2007, issued onAug. 31, 2010 as U.S. Pat. No. 7,784,544, which is incorporated byreference herein in its entirety.

This application is based upon U.S. Provisional Patent Application No.60/761,550, filed Jan. 24, 2006, and claims the benefit of the filingdate thereof.

BACKGROUND

The statements in this section merely provide background informationrelated to the present disclosure and may not constitute prior art.

This invention relates to the techniques used for stimulatinghydrocarbon-bearing formations—i.e., to increase the production ofoil/gas from the formation and more particularly, to a process foroptimizing fluids for and monitoring fluid rheological performanceduring fracture stimulation treatments.

Hydrocarbons (oil, natural gas, etc.) are obtained from a subterraneangeologic formation (i.e., a “reservoir”) by drilling a well thatpenetrates the hydrocarbon-bearing formation and thus causing a pressuregradient that forces the fluid to flow from the reservoir to the well.Often, a well production is limited by poor permeability either due tonaturally tight formations or due to formation damages typically arisingfrom prior well treatment, such as drilling, cleaning etc.

To increase the net permeability of a reservoir, it is common to performa well stimulation. A common stimulation technique is hydraulicallyfracturing a formation penetrated by a wellbore. Hydraulic fracturingtypically consists of pumping a proppant-free viscous fluid, or pad,usually water with some fluid additives to generate high viscosity, intoa well faster than the fluid can escape into the formation so that thepressure rises and the rock breaks, creating artificial fractures and/orenlarging existing fractures. Then, proppant particles are added to thefluid to form a slurry that is pumped into the fracture to prevent itfrom closing when the pumping pressure is released. The proppantsuspension and transport ability of the treatment base fluidtraditionally depends on the type and concentration of viscosifyingagent added.

Modern fracturing technology relies on fluids that exhibit flow behaviorthat changes over the course of a fracturing treatment. A fracturingfluid must be viscous enough to carry the proppant through theperforations and through the fracture, and to minimize fluid loss to theformation. On the other hand, the fluid should ideally be thinner in thetubing to limit horsepower requirements and to minimize sheardegradation. To facilitate an efficient clean-up, its viscosity must bereduced to an absolute minimum after the treatment is over, thusensuring optimum fracture conductivity and well productivity. Withmodern fracturing technology, a single fluid can meet all of theserequirements, but a successful fracturing treatment requires a carefulfluid design, i.e. the fluid composition should be chosen based onformation temperature and pressure, pumping rate, pumping time,completion capacity, water quality etc. The fluid design for afracturing treatment is traditionally determined based on bothexperience and laboratory testing, whereas modeling has previously onlyplayed a minor role or no role at all.

As stated above, for fracturing fluid designs, fluid compositions andbreaker schedules for a fracturing job is most commonly determined bylab rheology measurements and/or experience. Models are not solely usedin the determination of the gel loading, crosslinker concentration,breaker schedule etc. In some instances, fracturing simulators employingsimple models such as the power law model and the Cross model are usedto represent the fluid rheology. These models are regressed to therheology data for the actual fluid being pumped, so experimental data isneeded. None of these models account for the live chemistry offracturing fluids, i.e., the fact that crosslinks are dynamically formedand broken, as well as polymer linkages that are broken by thermaldegradation or degradation by oxidizers.

Further, the use of techniques to quality assure, quality control(QA/QC) the fracturing fluid real-time are not known or used at thistime. QA/QC of fluids typically are conducted on a time delayedsampling-testing basis.

Therefore, the need exists for methods that can reduce the number oflaboratory experiments needed, as well as techniques which enablereal-time QA/QC of fracturing fluids, so that the treatment may beadjusted if needed. Techniques which achieve the above would be highlydesirable, and these needs are met at least in part by the followinginvention.

SUMMARY OF THE INVENTION

In one aspect of the invention relates to techniques used for treatinghydrocarbon-bearing formations—such as to increase the production ofoil/gas from the formation and more particularly, to a process fortreating a subterranean formation by optimizing fluids for andmonitoring fluid placement during treatment.

In accordance with the invention, in the treatment method, a rheologymodel is used which enables prediction of fluid rheology propertiesduring a treatment operation, such as a fracturing treatment. Thefoundation of the model is an explicit description of the reactionchemistry, which describes how the number of crosslinks and brokenpolymer linkages develops in time under the influence of crosslinkers,breakers, and/or thermally induced effects and pressure effects. In oneaspect, when used as a tool for simulating the fluid viscosity, themodel can help optimizing the fluid design and optional breaker schedulefor a hydraulic fracturing treatment.

In some embodiments, the methods for treating a subterranean formationpenetrated by a wellbore, involve predicting a plurality of formationtreatment scenarios (fluid formulation and fluid delivery schedule) fora fluid comprising a polymer and a crosslinker. The model includesinputting a concentration of polymer and a concentration of crosslinker,inputting temperature and pressure and shear rate profiles over thecourse of the treatment, determining the concentration of crosslinksover the course of the treatment based upon the inputted values forconcentration of polymer, concentration of crosslinker, temperatureprofile over the treatment period, pressure profile over the treatmentperiod, and shear rate profile over the treatment period. The fluidrheology over the over the treatment period based on the concentrationof crosslinks over the course of the treatment is then calculated. Atreatment scenario which provides optimal fluid rheology propertiesduring the course of treatment and minimal fluid viscosity after thetreatment may then be selected, the fluid prepared and injected into thewellbore to treat the subterranean formation. For any of the embodimentsof the invention, concentration of breaker may optionally be consideredin the model as well, as well as other factors, for example, pHprofiles, changes in salinity, or any other factors which may affect therheology of the fluid, or fluids, over the course of the treatment.

In some other embodiments, the methods involve predicting a series offormation treatment scenarios for a fluid comprising a polymer and acrosslinker by inputting a concentration of polymer and a concentrationof crosslinker, inputting temperature and pressure and shear rateprofiles over the course of the treatment, determining the concentrationof crosslinks over the course of the treatment, and then calculating thefluid rheology over the treatment period based on the concentration ofcrosslinks. After which, a treatment scenario is selected which providesoptimal fluid rheology properties after completion of the treatment.

In yet other embodiments, the methods involve predicting a series offormation treatment scenarios for a linear fluid comprising a byinputting a concentration of polymer, inputting temperature and pressureand shear rate profiles over the course of the treatment, determiningthe fluid rheology over the course of the treatment by consideringconcentration of polymer, and temperature, pressure and shear rateprofiles over the course of the treatment. Thereafter, a treatmentscenario which provides optimal fluid rheology properties during thecourse of treatment may then be selected, the fluid prepared andinjected into the wellbore to treat the subterranean formation.

In some method embodiments of the invention, the fluid rheology isdetermined by coupling irreversible reactions of the crosslinker andpolymer with equilibrium reactions of the crosslinker and polymer as afunction of time, temperature, pressure, polymer concentration, andcrosslinker concentration.

Methods of the invention may be used for any subterranean formationtreatments, including but not limited to fracturing treatments,acidizing treatments, wellbore cleanout operations, gravel packingoperations, acid diversion treatments, and fluid loss controloperations. Some methods may be used in a hydraulic fracturing simulatorwhereby fluid rheology is calculated to simulate the fracture placementand an optimal pumping sequence is determined.

The selection of a treatment scenario may be based upon any desiredfactors such as, but not limited to, incorporating a minimum amount ofpolymer, maximizing rheology, optimized retained conductivity, optimizedclean-up, minimizing rheology or fraction, and the like.

In some embodiments, the methods can further include monitoring thestimulation treatment and evaluating the fluid performance based uponreal-time monitoring. In some of these embodiments values such as fluidpumping rate and fluid travel time to a treatment zone are input intothe model.

Any suitable polymers may be used in fluids used according to theinvention, including guar, hydropropyl guar (HPG), carboxymethyl guar(CMG), carboxymethylhydroxypropyl guar, cellulose, hydroxyethylcellulose(HEC), hydroxypropylcellulose (HPC), carboxymethylhydroxyethylcellulose(CMHEC), xanthan, diutan, whelan gum, polyacrylamide, polyacrylatepolymers. Preferable crosslinkers, when used, are zirconium, titanium orborate based crosslinkers. In some preferred instances, theconcentration of polymer ranges from about 15 to about 40 pounds perthousand gallons of treatment fluid, while in some other preferredembodiments the polymer is incorporated from about 1 to about 10 poundsper thousand gallons.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may be understood by reference to the followingdescription taken in conjunction with the accompanying drawings: FIG. 1show the viscosity reduced by the viscosity at complete crosslink as afunction of the crosslink concentration.

FIG. 1 illustrates the ratio of partially crosslinked fluid viscosity tofully crosslinked fluid viscosity as a function of the crosslinkconcentration.

FIG. 2 illustrates dimensionless crosslinking viscosity increment as afunction of the generalized shear rate where the transition betweenNewtonian and power law behavior occurs at a generalized shear rate of1.

FIG. 3 shows experimental and simulated rheology for a fluid containing20 ppt guar polymer and 3.1 gpt borate crosslinker solution, withoutbreaker.

FIG. 4 illustrates experimental and simulated rheology for a fluidcontaining 30 ppt guar polymer, 4.4 gpt borate crosslinker solution and2.2 ppt ammonium persulfate

FIG. 5 is a plot of experimental and simulated rheology for a fluidcontaining 24.9 ppt guar polymer, 2.2 gpt borate crosslinker solution,1.1 ppt ammonium persulfate and 0.5 gpt triethanolamine solution

FIG. 6 shows calculated versus experimental viscosities during a Fann 50experiment where the spikes correspond to the shear ramps, where theshear rate is ramped down from 100 s⁻¹ (base) to 25 s⁻¹ and back up tothe base shear rate.

FIG. 7 illustrates experimental and calculated viscosity profiles forthe Fann 50 experiment shown in FIG. 6.

FIG. 8 is a plot of calculated vs. experimental viscosities during alow-shear rheology experiment for a gel containing 30 ppt guar polymerand 3.6 gpt borate crosslinker solution at 125° F.

FIG. 9 is a plot of calculated vs. experimental viscosities profiles fora gel containing 30 ppt guar polymer, 3.6 gpt borate crosslinkersolution at 125° F.

DETAILED DESCRIPTION OF THE INVENTION

At the outset, it should be noted that in the development of any suchactual embodiment, numerous implementation—specific decisions must bemade to achieve the developer's specific goals, such as compliance withsystem related and business related constraints, which will vary fromone implementation to another. Moreover, it will be appreciated thatsuch a development effort might be complex and time consuming but wouldnevertheless be a routine undertaking for those of ordinary skill in theart having the benefit of this disclosure.

The description and examples are presented solely for the purpose ofillustrating the preferred embodiments of the invention and should notbe construed as a limitation to the scope and applicability of theinvention. While the compositions used in methods of the invention maybe described herein as comprising certain materials, it should beunderstood that the composition could optionally comprise two or morechemically different materials. In addition, the composition can alsocomprise some components other than those cited. In the summary of theinvention and this detailed description, each numerical value should beread once as modified by the term “about” (unless already expressly somodified), and then read again as not so modified unless otherwiseindicated in context. Also, in the summary of the invention and thisdetailed description, it should be understood that a concentration rangelisted or described as being useful, suitable, or the like, is intendedthat any and every concentration within the range, including the endpoints, is to be considered as having been stated. For example, “a rangeof from 1 to 10” is to be read as indicating each and every possiblenumber along the continuum between about 1 and about 10. Thus, even ifspecific data points within the range, or even no data points within therange, are explicitly identified or refer to only a few specific, it isto be understood that inventors appreciate and understand, that any andall data points within the range are to be considered to have beenspecified, and that inventors possession of the entire range and allpoints within the range.

As used herein, the phrases “fully crosslinked” and “completelycrosslinked” means a polymer/crosslinker system which has effectivelydeveloped the maximum practical viscosity under the particularconditions of use, unless indicted otherwise or clear from the contextwithin which the term is used. Unless otherwise indicated, the term“treatment scenario” means the fluid formulation and treatment schedulefor the fluid as used to treat a subterranean formation. The term“rheology” in the broadest sense of the term, that part of mechanicswhich deals with the relation between force and deformation in materialbodies. The nature of this relation depends on the material of which thebody is constituted. It is customary to represent the deformationbehavior of fluids by the model of the linear viscous or newtonian fluid(displaying the property known as viscosity). These classical modelsare, however, inadequate to depict certain nonlinear and time-dependentdeformation behavior that is sometimes observed. It is thesenonclassical behaviors which are to as rheological behavior, orrheology. Rheological behavior is particularly readily observed inmaterials containing polymer molecules which typically contain thousandsof atoms per molecule.

This invention relates to the techniques used for treatinghydrocarbon-bearing formations—such as to increase the production ofoil/gas from the formation and more particularly, to a process fortreating a subterranean formation by optimizing fluids for andmonitoring fluid placement during treatment. Subterranean formationtreatments include, but are not limited to, fracturing, acidizing,wellbore cleanout, gravel packing, acid diversion, fluid loss control,and the like. Methods of the invention employ a rheology model thatdirectly describes the chemical reactions that occur in a crosslinkedviscosifying agent based treatment fluid. One example of such a fluid isa borate-crosslinked guar-based fracturing fluid.

The methods of the invention are useful for subterranean formationtreatment, which also includes a wellbore penetrating the formation, andinclude such methods as treatment fluid design, breaker schedule design,rheology representation in treatment simulators, and even real-timeQA/QC of treatment fluid rheology. Preferably, the embodiments of theinvention are fracturing methods which include design of the fracturingfluid, design of the fracturing treatment, injection of the fracturingfluid into the wellbore, stimulating the formation, andmonitoring/optimizing the fluid/treatment based upon real-timemonitoring.

Methods of the invention incorporate models for such tasks as fluiddesign, breaker schedule design, rheology representation in treatmentsimulators, such as fracturing simulators, and the like. The novelty ofthe models is its direct link between the live fracturing fluidchemistry and its rheology. One aspect in the model is the conversionfrom the crosslink concentration, which is delivered by the chemistrypart of the model, to the viscosity. The viscosity (y) is related to thecrosslink concentration [GxB−Gx], mole/L (x) via an s-shapedy=c₁(1−sec(c₂x)) function, characterized by three regions. As thecrosslink concentration is gradually increased, the fluid passes throughthree distinct rheology regimes, as shown in FIG. 1, and as explainedbelow.

In the following, the characteristics of the three regimes will bedescribed along with molecular and electrostatic explanation for theviscosity behavior:

-   -   Regime 1: The weakly crosslinked regime. The limited attraction        caused by the low density of intermolecular crosslinks is easily        overcome by the shear stress, i.e., the molecules are separated        by the fluid flow and the viscosity increase therefore remains        low.    -   Regime 2: The transition region. The number of intermolecular        crosslinks increases and the number of overlaps that have        sufficiently high attraction energy to withstand the shear        stress exerted by the flow increases. The resistance to flow, or        the viscosity, therefore increases rapidly with the crosslink        density.    -   Regime 3: Most of the overlaps have sufficiently high density of        intermolecular crosslinks to resist the shear stress exerted by        the flow. A further increase of the crosslink density has only        limited effect on the fluid viscosity.

The target regime for a typical borate crosslinked guar-based fracturingfluids is regime 3, where the viscosity is fully developed and the fluidis robust, i.e., the viscosity is relatively insensitive to changes inthe crosslink concentration.

While some fluids used in treatment methods of the invention areborate-crosslinked guar-based fracturing fluids, the fluids may be anycrosslinked polymer based fluids, or linear polymer based fluids, usedfor treating a subterranean formation. The fluids typically include apolymer viscosifying agent and a crosslinker. Non-limiting examples ofpolymer viscosifiers include guar gums, high-molecular weightpolysaccharides composed of mannose and galactose sugars, or guarderivatives such as hydropropyl guar (HPG), carboxymethyl guar (CMG),and carboxymethylhydroxypropyl guar (CMHPG). Cellulose derivatives suchas hydroxyethylcellulose (HEC) or hydroxypropylcellulose (HPC) andcarboxymethylhydroxyethylcellulose (CMHEC) may also be used. Any usefulpolymer may be used in either crosslinked form, or without crosslinkerin linear form. Xanthan, diutan, and scleroglucan, three biopolymers,have been shown to be useful as viscosifying agents. Synthetic polymerssuch as, but not limited to, polyacrylamide and polyacrylate polymersand copolymers are used typically for high-temperature applications.Also, associative polymers for which viscosity properties are enhancedby suitable surfactants and hydrophobically modified polymers can beused, such as cases where a charged polymer in the presence of asurfactant having a charge that is opposite to that of the chargedpolymer, the surfactant being capable of forming an ion-pair associationwith the polymer resulting in a hydrophobically modified polymer havinga plurality of hydrophobic groups, as described published U.S. Pat. App.No. US 2004209780, Harris et. al.

When incorporated, the polymer viscosifier may be present at anysuitable concentration. In various embodiments hereof, the viscosifyingagent can be present in an amount of up to less than about 60 pounds perthousand gallons of liquid phase, or from about 15 to less than about 40pounds per thousand gallons, from about 15 to about 35 pounds perthousand gallons, 15 to about 25 pounds per thousand gallons, or evenfrom about 17 to about 22 pounds per thousand gallons. Generally, theviscosifying agent can be present in an amount of from about 1 to lessthan about 50 pounds per thousand gallons of liquid phase, with a lowerlimit of polymer being no less than about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, 12, 13, 14, 15, 16, 17, 18, or 19 pounds per thousand gallons of theliquid phase, and the upper limited being less than about 50 pounds perthousand gallons, no greater than 59, 54, 49, 44, 39, 34, 30, 29, 28,27, 26, 25, 24, 23, 22, 21, or 20 pounds per thousand gallons of theliquid phase. In some embodiments, the polymers can be present in anamount of about 20 pounds per thousand gallons. Hydroxypropyl guar,carboxymethyl hydroxypropyl guar, carboxymethyl guar, cationicfunctional guar, guar or mixtures thereof, are preferred polymers foruse herein as a gelling agent. Fluids incorporating polymer viscosifiersmay have any suitable viscosity depending upon the particular needs of agiven operation. For many operations, the fluids preferably have aviscosity value of about 50 mPa-s or greater at a shear rate of about100 s⁻¹ at treatment temperature, more preferably about 75 mPa-s orgreater at a shear rate of about 100 s⁻¹, and even more preferably about100 mPa-s or greater. In the case of a slickwater fracturing, alsocommonly referred to as a water fracture operation, the fluid may havesuitably low, but effective, viscosity values, and low polymer loadings,preferably less than about 15 pounds per thousand gallons, morepreferably from about 1 to about 10 pounds per thousand gallons.

Incorporating crosslinkers into the fluids further augments theviscosity of the treatment fluid. Crosslinking consists of theattachment of two polymeric chains through the chemical association ofsuch chains to a common element or chemical group, whereas such elementor group is referred to as the crosslinker. Typical crosslinkers arepolyvalent metal ions, more often zirconium or titanium ions, or borateions. Crosslinking is very sensitive to the prevailing pH. For example,crosslinking with borate ions can be performed only in alkaline media(pH>8). pH-regulating systems (“buffers”) are often required to achieveeffective crosslinking with metal ions.

Fluids used in the invention may be based upon and aqueous or nonaqueousmedium, or even gelled oil. When the fluid is based upon an aqueousmedium, the medium may be water or brine. In those embodiments of theinvention where the aqueous medium is a brine, the brine is watercomprising inorganic salts and/or organic salts. Preferred inorganicsalts include alkali metal halides, more preferably potassium chloride.The carrier brine phase may also comprise an organic salt morepreferably sodium or potassium formate. Preferred inorganic divalentsalts include calcium halides, more preferably calcium chloride orcalcium bromide. Sodium bromide, potassium bromide, or cesium bromidemay also be used. The salt is chosen for compatibility reasons i.e.where the reservoir drilling fluid used a particular brine phase and thecompletion/clean up fluid brine phase is chosen to have the same brinephase.

A fiber component may be included in the fluids of the invention toachieve a variety of properties including improving particle suspension,and particle transport capabilities, and gas phase stability. Fibersused may be hydrophilic or hydrophobic in nature, but hydrophilic fibersare preferred. Fibers can be any fibrous material, such as, but notnecessarily limited to, natural organic fibers, comminuted plantmaterials, synthetic polymer fibers (by non-limiting example polyester,polyaramide, polyamide, novoloid or a novoloid-type polymer),fibrillated synthetic organic fibers, ceramic fibers, inorganic fibers,metal fibers, metal filaments, carbon fibers, glass fibers, ceramicfibers, natural polymer fibers, and any mixtures thereof. Particularlyuseful fibers are polyester fibers coated to be highly hydrophilic, suchas, but not limited to, DACRON® polyethylene terephthalate (PET) Fibersavailable from Invista Corp. Wichita, Kans., USA, 67220. Other examplesof useful fibers include, but are not limited to, polylactic acidpolyester fibers, polyglycolic acid polyester fibers, polyvinyl alcoholfibers, and the like. When used in fluids of the invention, the fibercomponent may be include at concentrations from about 1 to about 15grams per liter of the liquid phase of the fluid, preferably theconcentration of fibers are from about 2 to about 12 grams per liter ofliquid, and more preferably from about 2 to about 10 grams per liter ofliquid.

Fluids used in accordance with the invention may also comprise abreaker. The purpose of this component is to “break” or diminish theviscosity of the fluid so that this fluid is more easily recovered fromthe formation during cleanup. With regard to breaking down viscosity,oxidizers, enzymes, or acids may be used. Breakers reduce the polymer'smolecular weight by the action of an acid, an oxidizer, an enzyme, orsome combination of these on the polymer itself. In the case ofborate-crosslinked gels, increasing the pH and therefore increasing theeffective concentration of the active crosslinker, the borate anion,reversibly create the borate crosslinks. Lowering the pH can just aseasily eliminate the borate/polymer bonds. At a high pH above 8, theborate ion exists and is available to crosslink and cause gelling. Atlower pH, the borate is tied up by hydrogen and is not available forcrosslinking, thus gelation caused by borate ion is reversible.

Fluids used in methods of the invention may further contain otheradditives and chemicals that are known to be commonly used in oilfieldapplications by those skilled in the art. These include, but are notnecessarily limited to, materials such as surfactants, foaming agents,crosslinking delay agent, breaker delay agents, particles, proppants,gas component, breaker aids, oxygen scavengers, alcohols, scaleinhibitors, corrosion inhibitors, fluid-loss additives, bactericides,friction reducers, latexes, emulsions, emulsifiers, and the like.

FIG. 1 illustrates the ratio of partially crosslinked fluid viscosity tofully crosslinked fluid viscosity as a function of the crosslinkconcentration. The method of obtaining the crosslink density isdescribed herein below. The transition between low and high viscosityoccurs in the crosslink concentration range from about 5×10⁻⁹ to about5×10⁻⁸ mole/L, so a fracturing fluid, for example, may be designed tohave a concentration equal to or greater than about 5×10⁻⁸ mole/L tohave fully developed viscosity, given the bottom hole and pumpingconditions. The scatter of the data at the plateau is caused in byexperimental uncertainty and, in part, by the simplified method used tocalculate the viscosity at full crosslink.

Harris (P. C. Harris. Chemistry and rheology of borate-crosslinkedfluids at temperatures to 300° F. Society of Petroleum Engineers, SPE24339:285-293, 1992, and P. C. Harris. Chemistry and rheology ofborate-crosslinked fluids at temperatures 300° F. Journal of PetroleumTechnology, March: 264-269, 1993) performed a qualitative study of howthe borate equilibrium affects the crosslinking chemistry. Mesmer et al.(R. E. Mesmer, C. F. Baes, and F. H. Sweeton. Acidity measurements atelevated temperatures. VI. Boric acid equilibria. Inorganic chemistry,11(3):537-543, 1971) studied the boric acid equilibria and provided andrelated the dissociation quotient to the ionic strength and temperature.More recently, Pezron et al. (E. Pezron, A. Ricard, F. Lafuma, and R.Audebert. Reversible gel formation induced by ion complexation. 1.borax-galactomannan interactions. Macromolecules, 21:1121-1125, 1988)and Bishop et al. (M. Bishop, N. Shahid, J. Yang, and A. R. Barron.Determination of the mode and efficacy of the cross-linking of guar byborate using MAS 11B NMR of borate cross-linked guar in combination withsolution 11B NMR of model systems. Dalton Trans., pages 2621-2634, 2004)studied the guar-borate reactions by NMR-spectropscopy and quantifiedthe equilibrium quotient of the crosslinking reactions. Theseequilibrium quotients can be coupled to provide an evaluation of thecrosslink density, which is among the main determinants of the fluidrheology. The result is a model that enables a prediction of how theviscosity may depend on the overall gel composition and its exposurehistory. A model is show as follows, as well as the performance of themodel. The above references are incorporated by reference thereto.

One embodiment of a model useful in the invention separates theviscosity into contributions from the solvent (s), the linear guar (lg)and crosslinking (xl):η=η^(s)+η^(lg)+η^(x1)  (1)This functionality ensures that the model reduces to the correct limitswhen crosslinking and/or gel concentration vanishes.

The solvent viscosity is calculated from a correlation that is fitted towater viscosity data from the CRC handbook (Lide, 2005):

$\begin{matrix}{{\ln\;\eta^{s}} = {{- 3.51} + \frac{500}{T - 151}}} & (2)\end{matrix}$where T is the absolute temperature in K and ηs is the solvent viscosityin cP. The pressure dependence of the solvent (i.e. water) viscosity issmall compared to the crosslinked gel and is not considered here.

The linear-gel viscosity is calculated by the Ellis-type model:

$\begin{matrix}{\frac{\eta_{0}^{\lg}}{\eta^{\lg}} = {1 + \frac{1}{285\;\beta^{- 0.76}}}} & (3)\end{matrix}$where η₀ ^(lg) is the zero-shear viscosity of the linear guar and β isthe so-called generalized shear rate:

$\begin{matrix}{\beta = {{\overset{.}{\gamma}\left( {\eta_{0}^{\lg} - \eta^{s}} \right)}\frac{M}{cRT}}} & (4)\end{matrix}$where, in turn, {dot over (γ)} is the shear rate, M is the polymermolecular weight, c is the gel concentration (mass/volume) and R is theuniversal gas constant. The zero-shear viscosity is calculated by amethod which relates the specific viscosity to the product between thehydrodynamic volume (intrinsic viscosity) and the gel concentrationthrough:

$\begin{matrix}{{\ln\frac{\eta_{0}^{\lg} - \eta^{s}}{\eta^{s}}} = \left\{ \begin{matrix}{0.429 + {1.61\;{\ln\left( {c\lbrack\eta\rbrack} \right)}}} & {{{for}\mspace{14mu}{c\lbrack\eta\rbrack}} < 3.37} \\{{- 3.27} + {4.66\;{\ln\left( {c\lbrack\eta\rbrack} \right)}}} & {{{for}\mspace{14mu}{c\lbrack\eta\rbrack}} \geq 3.37}\end{matrix} \right.} & (5)\end{matrix}$

This function is continuous but not smooth. The physical explanation forthis is that the concentration dependence is stronger above the overlapconcentration than below. The intrinsic viscosity is calculated by theMark-Houwink equation:[η]=KM^(a)  (6)with the following temperature dependent parameters:

$\begin{matrix}{{\ln\; K} = {{- 6.05} - \frac{1.29 \cdot 10^{3}}{T}}} & (7) \\{\alpha = {0.55 + \frac{1.04 \cdot 10^{2}}{T}}} & (8)\end{matrix}$where the temperature is in K and the intrinsic viscosity in dl/g. Thepressure dependence of the linear guar viscosity is small compared tothe crosslinked gel is not considered here.

The fluid viscosity related to crosslinking is established by firstcalculating the chemical contribution at a reference shear rate,followed by a shear-rate correction by an Ellis-type rheology model. 100s⁻¹ is picked here as the reference shear rate. The chemicalcontribution at the reference shear rate is given by:

$\begin{matrix}{\eta_{ref}^{xl} = {Q_{1}{c\lbrack\eta\rbrack}\left( {1 - {{sech}\left( {1.360\frac{\lbrack{XL}\rbrack}{\lbrack{XL}\rbrack_{trans}}} \right)}} \right)}} & (9)\end{matrix}$where Q₁ is an empirical parameter and [XL]trans is the crosslinkconcentration at which the viscosity is half of that at completecrosslinking. The parameters were regressed to experimental viscositydata and are listed in table 1.

TABLE 1 Parameters in the rheology model. Parameter Value Unit Relatedto Q₁ 1.173E+02 cP Viscosity calculation, equation 10 [XL]_(trans)1.539E−08 mole/L H^(II) −2.080E+04   J/mole 1st complexation reaction.Equilibrium constant given S^(II) −6.050E+01   J/mole/K by equation 23H^(III) −2.060E+04   J/mole Crosslinking. Equilibrium constant given byequation S^(III) −6.800E+01   J/mole/K 24 k₀ ^(IV) 4.652E+02 min⁻¹Thermal degradation of the polymer linkages. Reaction E_(a) ^(IV)6.372E+04 J/mole rate given by equation 26 k₁ ^(IV) 6.015E+12 L/mole k₀^(V) 5.587E+05 min⁻¹ Gradual pH reduction modelled as acid release. AcidE_(a) ^(V) 2.038E+04 J/mole release rate given by equation 27 c_(HAc)1.173E−04 mole/g k₀ ^(VII) 4.297E+16 min⁻¹ Thermally induced persulfatedecomposition. Reaction E_(a) ^(VII) 1.249E+05 J/mole rate given byequation 28. k₀ ^(VIII) 8.142E+04 L/mole/min Persulfate decompositioncatalyzed by triethylamine. E_(a) ^(VIII) 1.325E+04 J/mole Reaction rategiven by equation 29 k₀ ^(IX) 3.253E+45 L/mole/min Break of the polymerlinkages caused by persulfate E_(a) ^(IX) 3.003E+05 J/mole radicals.Reaction rate given by equation 30

Although the functionality of equation 9 is empirical, there is strongevidence that the gel viscosity first increases dramatically thenstabilizes when gradually increasing the crosslink concentration. Thisis well described by the 1−sech(f([XL])) functionality, but a furtherimproved scaling with temperature and gel loading is achieved whenintroducing the product c[η] in equation 9. It is seen from equation 9that η^(xl) _(ref) approaches Q₁c[η] asymptotically when the crosslinkconcentration exceeds the transition crosslink concentration. The actualcomposition, and thereby also the concentration of crosslinks, is acomplex function of the overall composition (gel load, crosslinkerconcentration, breaker concentration, etc.), temperature and time, andis governed by a set of chemical reactions.

In one embodiment, the chemistry of borate-crosslinked guar-based fluidsis represented by a simple model that considers the crosslinking-relatedreactions to be at equilibrium. The key constituents are assumed to beH⁺, OH⁻, boric acid B(OH)₃, borate B(OH)₄ ⁻, crosslink sites located onthe galactose groups of the polymer G_(x), connections between crosslinksites formed by borate (crosslinks) G_(x)B⁻G_(x), The equilibriumdistribution between these components can be calculated by a GibbsEnergy Minimization, as described by Smith and Missen, Chemical ReactionEquilibrium Analysis: Theory and Algorithms. Krieger, 1982, incorporatedherein by reference thereto, or by solving a set of equilibrium quotientequations coupled with element balances. An independent set of chemicalreactions that equilibrates the assumed instantaneously equilibratingspecies is:Reaction I: B(OH)₃+OH⁻

B(OH)₄ ⁻(═B⁻)  (10)Reaction II: B⁻+G_(x)

B ⁻G_(x)  (11)Reaction III: G_(x)+B⁻G_(x)

G_(x)B⁻G_(x)(═XL)  (12)

The gel degrading reactions, on the other hand, occur relatively slowlyand are therefore modeled by kinetic reactions. Gel degradation occursrapidly in the presence of breakers, but the gel also degrades bythermally induced hydrolysis, which breaks the acetal linkages betweenthe mannose units (backbone) and between the mannose and galactose sidechains. Denoting an acetal linkage by LL, we describe this effect by thereaction:Reaction IV: LL→2L  (13)

In addition to reaction (4), some viscosity loss may be attributed to agradual reduction of the pH. In the framework of the model, this isdescribed by considering the guar to contain an organic acid that isgradually released to the solution:Reaction V GHAc→G+HAc  (14)

The organic acid may then displace the borate equilibria, equation 10,towards boric acid by consuming hydroxyl ions:Reaction VI HAc+OH⁻

Ac⁻+H₂O  (15)

Finally, the gel degradation caused by persulfate breakers occurs in twostages, first by persulfate decomposition, then by an attack of sulfateradicals on the polymer linkages. At high temperatures, the persulfatedecomposes into two sulfate radicals:Reaction VII: SS→2S  (16)

At low temperatures, this reaction is slow, and breaker aid may beadded. The breaker aid is a solution of triethanol amine (TEA), whichacts as a catalyst for the decomposition:Reaction VIII: SS+TEA→2S+TEA  (17)

The sulfate radicals generated in reaction VII and VIII then attack theacetal bonds between the sugar units in the polymer:Reaction IX: S+LL→SL+L  (18)

In this reaction, either SL or L may be a radical, which is in realitycapable of attacking another LL, creating another radical, etc. Thischain reaction will propagate until termination, which occurs when tworadicals meet and react. The propagation and termination of the chainreaction has been omitted in the current version of the model, whichtherefore lumps the entire breaker effect into reactions (VII), (VIII)and (IX). During gel degradation, the number-average molecular weightcan be calculated as the total weight divided by the total number ofmolecules, which gives:

$\begin{matrix}{M^{x\; 1} = {\frac{n_{0}^{\lg}M^{\lg}}{n_{0}^{\lg} + \frac{nLL}{2} - n_{x\; 1}} = \frac{c_{0}M^{\lg}}{c_{\lg} + \frac{\lbrack{LL}\rbrack}{2} - \lbrack{XL}\rbrack}}} & (19)\end{matrix}$where n is the mole number, c is the molar nominal concentration andsubscript 0 indicates that the quantity refers to the linear, non-brokenpolymer. This formula assumes that all crosslinks are intermolecular.The total number of molecules in the denominator appears by realizingthat each (intermolecular) crosslink reduces the number of polymersegments by one and that each broken acetal linkage LL increases thenumber of polymer segments by one. The change in the molecular weightcaused by crosslinking and gel degradation thus affects η^(xl) _(ref)through [XL] and [η] (see equation 9). The actual composition, includingthe crosslink concentration, then becomes a time-dependent functionalgoverned by the array of component chemical potentials and reactionrates for reactions I through IX. The equilibrium factors are given by:

$\begin{matrix}{K^{I} = \frac{\left\lbrack B^{-} \right\rbrack}{\left\lbrack {B({OH})}_{3} \right\rbrack\left\lbrack {OH}^{-} \right\rbrack}} & (20) \\{K^{II} = \frac{\left\lbrack {G_{x}B^{-}} \right\rbrack}{\left\lbrack G_{x} \right\rbrack\left\lbrack B^{-} \right\rbrack}} & (21) \\{K^{III} = \frac{\lbrack{XL}\rbrack}{\left\lbrack {G_{x}B^{-}} \right\rbrack\left\lbrack G_{x} \right\rbrack}} & (22) \\{K^{VI} = {\frac{\left\lbrack {Ac}^{-} \right\rbrack}{\lbrack{HAc}\rbrack\left\lbrack {OH}^{-} \right\rbrack} = \frac{K_{a,{HAc}}}{K_{a,{H\; 2O}}}}} & (23)\end{matrix}$where K_(a, HAc) and K_(a, H2O) are the dissociation constants for HAcand water, respectively. The equilibrium constants are related to thereaction entropies and enthalpies through the thermodynamicrelationship:G=−RT ln K  (24)G=H−TS  (25)where G is the reaction free energy, H is the reaction enthalpy and S isthe reaction entropy.

The chemical potential for the crosslink complex has beenback-calculated from equilibrium factors given by M. Bishop, N. Shahid,J. Yang, and A. R. Barron. Determination of the mode and efficacy of thecross-linking of guar by borate using MAS 11B NMR of borate cross-linkedguar in combination with solution 11B NMR of model systems, DaltonTrans., pages 2621-2634, 2004. in terms of reaction enthalpy andentropy, incorporated herein by reference thereto. The values used aresummarized in table 1.

The rates of the reactions related to the gel degradation are assumed tobe first order in each of the reactants, and are described byArrhenius-type expressions:

$\begin{matrix}{R^{IV} = {k_{0}^{IV}{\exp\left( {- \frac{E_{a}^{IV}}{RT}} \right)}\frac{\lbrack{LL}\rbrack}{1 + {k_{1}^{IV}\lbrack{TEA}\rbrack}}}} & (26) \\{R^{V} = {k_{0}^{V}{{\exp\left( {- \frac{E_{a}^{V}}{RT}} \right)}\lbrack{GHAc}\rbrack}}} & (27) \\{R^{VII} = {k_{0}^{VII}{{\exp\left( {- \frac{E_{a}^{VII}}{RT}} \right)}\lbrack{SS}\rbrack}}} & (28) \\{R^{VIII} = {k_{0}^{VIII}{{{\exp\left( {- \frac{E_{a}^{VIII}}{RT}} \right)}\lbrack{SS}\rbrack}\lbrack{TEA}\rbrack}}} & (29) \\{R^{IX} = {k_{0}^{IX}{{{\exp\left( {- \frac{E_{a}^{IX}}{RT}} \right)}\lbrack S\rbrack}\lbrack{LL}\rbrack}}} & (30)\end{matrix}$where k₀ is the pre-exponential (frequency) factor and E_(a) is theactivation energy for each reaction. Reaction IV has been modified toaccount for the stabilizing effect TEA has on the gel. Thetime-dependence of the composition at constant volume (incompressiblefluid) is then given by:

$\begin{matrix}{\frac{\mathbb{d}\lbrack{SS}\rbrack}{\mathbb{d}t} = {- \left( {R^{VII} + R^{VIII}} \right)}} & (31) \\{\frac{\mathbb{d}\lbrack S\rbrack}{\mathbb{d}t} = {\left( {R^{VII} + R^{VIII}} \right) - R^{IX}}} & (32) \\{\frac{\mathbb{d}\lbrack{LL}\rbrack}{\mathbb{d}t} = {- \left( {R^{IV} + R^{IX}} \right)}} & (33) \\{\frac{\mathbb{d}\lbrack{GHAc}\rbrack}{\mathbb{d}t} = {- R^{V}}} & (34)\end{matrix}$

The integration of the irreversible reactions is coupled with theequilibrium reactions to provide the actual fluid composition, includingthe crosslink concentration, as a function of time, temperature andpressure which is subsequently used to calculate the crosslinked gelviscosity.

The shear-dependence of the crosslinking contribution to the viscosityis described by a molecular version of the Eyring model:{dot over (γ)}=A sin h(Bτ)  (35)where {dot over (γ)} is the shear rate and τ is the shear stress. A andB are related to molecular parameters through:

$\begin{matrix}{A = {2\frac{kT}{h}{\exp\left( {- \frac{E_{att}}{kT}} \right)}}} & (36) \\{B = \frac{a^{3}}{2\;{kT}}} & (37)\end{matrix}$where E_(att) is the attractive energy between two polymer coils, a isthe distance between two polymer coil centers, k is Boltzmann's constantand h is Planck's constant. The attraction is dominated by theentanglements and crosslinks that are formed in the overlap regionsbetween different polymer chains. The viscosity fix-points at thereference shear rate provided by equation 9 is matched by the followingconstraint:

$\begin{matrix}{A = \frac{{\overset{.}{\gamma}}_{ref}}{\sinh\left( {B\;\tau_{ref}} \right)}} & (38)\end{matrix}$The distance between two polymer coil centers a is calculated from thegel concentration, i.e., the shear model contains no adjustableparameters.

In dimensionless form, the Eyring model can be formulated as:

$\begin{matrix}{\overset{\_}{\eta} = {\frac{\eta}{\eta_{0}} = \frac{\ln\left( {\beta + \sqrt{\beta^{2} + 1}} \right)}{\beta}}} & (39)\end{matrix}$According to this model, the fluid viscosity is Newtonian at low shearrates and the slope of the curve slowly approaches −1 at high shearrates. The model parameters obtained for a model that uses 100 s⁻¹ asthe reference shear rate are listed in table 1.

Formation and downhole pressure and temperature can have an impact onfluid rheology. In the case of pressure, when there is adequate pressurepresent in the treatment or delivery environment, the effectivecrosslinking functionality of a crosslinking agent, such as a borate,may be significantly reduced. Such pressures are those on the order ofmagnitude of 10³ psi or greater, such 4×10³ psi or greater. At 4×10³psi, it has been unexpectedly discovered that measured viscosity isabout half of the viscosity of a borate crosslinker at ambient surfacepressure. Thus, the pressure affects on a borate crosslinker can betaken into account in some embodiments of the invention.

Methods of the invention may also be used to for real-time QA/QC of thefluids, thus making possible to adjust the fluid components during anoperation to achieve a further optimized fluid and treatment schedule.The rheology model can be used to further extrapolate monitored surfacecharacteristics such as viscosity, pumping rate, temperature, polymerconcentration, crosslinker concentration, breaker concentration tobottomhole conditions

The following example is presented to illustrate the preparation andproperties of energized aqueous fluids comprising heteropolysaccharidesand a surfactant, and should not be construed to limit the scope of theinvention, unless otherwise expressly indicated in the appended claims.All percentages, concentrations, ratios, parts, etc. are by weightunless otherwise noted or apparent from the context of their use.

Example

The following example illustrates the methods of the invention, asdescribed in the preferred embodiments.

Viscosity predictions at reference shear rate (100 s⁻¹) are made. Forsystems containing no breaker, the model was examined against 68 Fann 50experiments for gels containing 15-40 ppt guar, 0.5-10.5 gpt of a 20% byweight aqueous solution of borax, and no breaker. 51 of these (84%) aresuccessfully represented by the model. For these analyses, successcriterion required that at viscosities lower than 250 mPa-s, predictionmust be within 50 mPa-s of experimental data, while at higherviscosities, prediction within 20% of experimental data. The successcriterion is less strict at low viscosities because of higherexperimental uncertainty of the Farm 50. A performance example is shownin FIG. 3.

For systems containing breaker, the model was examined against 42 Fann50 experiments containing 15-40 ppt guar polymer, 1.1-4.4 gpt of a 20%by weight aqueous solution of borax, 0.6-2.2 ppt ammonium persulfatebreaker, and 0.3-1.0 gpt triethanolamine. 35 of these (83%) aresuccessfully represented by the model. Performance examples at high- andlow temperature are shown in FIGS. 4 and 5.

For shear dependence correlation trends, the model was examined against188 Fann 50 shear ramps for non-broken gels. These ramps are measuredduring a standard Fann 50 experiment, where the shear rate is rampeddown from the base shear rate 100 s⁻¹ to 25 s⁻¹ and back up to the baseshear rate. This technique can be found in API Recommended Practice 39,entitled “Recommended Practices on Measuring the Viscous Properties of aCross-linked Water-based Fracturing Fluid”, dated May 1998. The Eyringmodel can not predict shear thickening, so ramps indicating shearthickening were excluded from this investigation. The average deviationbetween calculated and experimental viscosities was 80 mPa-s. Aperformance example is shown in FIGS. 6 and 7. At lower shear rates themodel over predicts the viscosities, but this can be corrected for byintroducing an empirical correction to the polymer-spacing parameter a,as shown in FIGS. 8 and 9.

These examples illustrate that a rheology model is enabled for boratecrosslinked guar-based fracturing fluids. The model enables a predictionof the viscosity during a fracturing job. The foundation of the model isan explicit description of the reaction chemistry, which describes howthe number of crosslinks and broken polymer linkages develops in timeunder the influence of crosslinkers, breakers and thermally inducedeffects. The viscosity predictions were successful for at least 84% ofthe data series investigated here. The average deviation betweencalculated and experimental viscosities was 80 mPa-s for 188 shear rampsbetween 100 and 25 When used as a tool for simulating the fluidviscosity, the model can help optimizing the fluid design and breakerschedule for a hydraulic fracturing treatment.

The particular embodiments disclosed above are illustrative only, as theinvention may be modified and practiced in different but equivalentmanners apparent to those skilled in the art having the benefit of theteachings herein. Furthermore, no limitations are intended to thedetails of construction or design herein shown, other than as describedin the claims below. It is therefore evident that the particularembodiments disclosed above may be altered or modified and all suchvariations are considered within the scope and spirit of the invention.Accordingly, the protection sought herein is as set forth in the claimsbelow.

1. A method for treating a subterranean formation, the methodcomprising: a. calculating a formation treatment plan comprising using atreatment fluid comprising a polymer and a crosslinker using a fluidrheology model comprising: i. inputting a concentration of polymer and aconcentration of crosslinker, ii. inputting temperature, formationpressure, and shear rate profiles over the course of the treatment, iii.determining a fluid rheology based upon the input values from i, and iiover the course of the treatment; b. implementing the formationtreatment plan comprising preparing and injecting the treatment fluidinto a wellbore; and c. treating the subterranean formation.
 2. Themethod of claim 1, further comprising inputting a concentration ofbreaker.
 3. The method of claim 1, further comprising inputting theconcentration of one or more pH modifiers over the course of thetreatment.
 4. The method of claim 1, wherein in the determining thefluid rheology is based upon incorporating a concentration of polymerand optimal clean-up after treatment.
 5. The method of claim 1, furthercomprising inputting a concentration of a crosslinking delay agent overthe course of the treatment.
 6. The method of claim 1, furthercomprising monitoring the treatment fluid and evaluating the fluidperformance downhole based upon real-time monitoring of the fluidcomposition at the surface.
 7. The method of claim 1, further comprisinginputting fluid pumping rate, wellbore configuration, and fluid traveltime to a treatment zone.
 8. The method of claim 1, wherein theformation treatment plan comprises fracturing a subterranean formation.9. The method of claim 1, wherein the formation treatment plan comprisesmonitoring fluid placement during a treatment selected from the groupconsisting of fracturing treatments, acidizing treatments, wellborecleanout operations, gravel packing operations, acid diversiontreatments, and fluid loss control operations.
 10. The method of claim1, wherein the polymer is selected from the group consisting of guar,hydropropyl guar (HPG), carboxymethyl guar (CMG),carboxymethylhydroxypropyl guar, cellulose, hydroxyethylcellulose (HEC),hydroxypropylcellulose (HPC), carboxymethylhydroxyethylcellulose(CMHEC), xanthan, diutan, whelan gum, polyacrylamide, polyacrylatepolymers, and wherein the crosslinker is a zirconium, titanium or boratebased crosslinker.
 11. The method of claim 1, wherein the polymer isguar and the crosslinker is a borate based crosslinker.
 12. The methodof claim 1, further comprising considering pressure effects on aborate-based crosslinker in determining the fluid rheology.
 13. Themethod of claim 1, wherein the fluid further comprises a proppant, andwherein the calculating a plan comprises inputting a concentration ofproppant.
 14. The method of claim 1, wherein a concentration of polymerranges from about 15 to about 40 pounds per thousand gallons.
 15. Themethod of claim 1, wherein the concentration of polymer is about 15pounds per thousand gallons or less.
 16. A method for treating asubterranean formation, the method comprising: a. calculating aformation treatment plan comprising using a treatment fluid comprising apolymer and a crosslinker using a fluid rheology model comprising: i. aconcentration of polymer and a concentration of crosslinker, ii.temperature, formation pressure, and shear rate profiles over the courseof the treatment, iii. a fluid rheology estimate based upon input valuesfrom i, and ii; b. implementing the formation treatment plan to form andintroduce the treatment fluid into a wellbore; and c. treating thesubterranean formation.
 17. The method of claim 16, wherein the modelfurther comprises a concentration of breaker.
 18. The method of claim16, wherein the model further comprises a concentration of one or morepH modifiers over the course of the treatment.
 19. The method of claim16, wherein the model further comprises a concentration of acrosslinking delay agent over the course of the treatment.
 20. Themethod of claim 16, further comprising monitoring the treatment fluidand evaluating the fluid performance downhole based upon real-timemonitoring of the fluid composition at the surface.
 21. A method fortreating a subterranean formation, the method comprising: a. calculatinga formation treatment plan comprising using a treatment fluid comprisinga polymer and a crosslinker using a fluid rheology model comprising: i.inputting a concentration of polymer, a concentration of crosslinker, aconcentration of breaker, concentration of one or more pH modifiers, anda crosslinking delay agent, ii. inputting temperature, formationpressure, and shear rate profiles over the course of the treatment, iii.determining a fluid rheology based upon the input values from i, and iiover the course of the treatment; and b. implementing the formationtreatment plan comprising preparing and injecting the treatment fluidinto a wellbore.